Poisson algebra ($\{fg,h\}=f\{g,h\}+g\{f,h\}$)

A Poisson algebra is a commutative associative algebra with a Lie bracket that is a derivation in each argument. Smooth functions on a symplectic manifold are its standard example. The linear span of moment-map components may form a finite-dimensional Lie subalgebra, while closing those components under multiplication normally produces a much larger Poisson algebra.

Let $h$ be central and a non-zero-divisor in an associative algebra, with commutative quotient modulo $h$. Dividing commutators by $h$ and reducing modulo $h$ is independent of the lifts, since changing a lift by $hc$ changes the quotient by a commutator, zero modulo $h$. Antisymmetry, the Leibniz rule and the Jacobi identity descend from the corresponding commutator identities. For the order filtration of a Weyl algebra, the convention is $\{{\xi }_i,x_j\}={\delta }_{ij}$, with the opposite sign for $\{x_i,{\xi }_j\}$.